For VCC=5V, ICSAT=5mA and the transistor manufacturers guaranteed βmin=100, find values for RC and RB that will ensure that the transistor just saturates when driven by VIN=+5V. Would it be safe to use this value in a mass-produced version of the circuit?

I calculated Rc and Rb using Icsat=(Vcc-Vcesat)/Rc
and Ibsat=(Vin-Vbesat)/Rb.
I got Rc=960 Ω
Rb=85 kΩ

I'm not very sure about the second part of the question. For mass-produced versions of the circuit does βmin vary? Is it safe to use these resistor values?

I'm not very sure about the second part of the question. For mass-produced versions of the circuit does βmin vary? Is it safe to use these resistor values?

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βmin does not vary so you can use this values for mass-production but you will be on the edge between saturation and active region. A more safe circuit, which assumes to work in saturation for mass production, is to assume that βmin is ten times less than the value noted on the datasheet and make your calculations again with this divided by 10 value. This is called hard saturation because you give the base 10 times the current it needs to saturate the transistor. However, take care not to exceed the maximum base current rating.